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Strong monodromy conjecture for defining polynomials of reduced projective curves having only weighted homogeneous singularities

Morihiko Saito

Abstract

Let $f$ be a defining polynomial of a reduced projective curve $C\subset{\bf P}^2$ having only weighted homogeneous singularities. We show that the strong monodromy conjecture for $f$ follows rather easily from arxiv:1609.04801v11 using a formula of Denef and Loeser for the Newton-nondegenerate case with three variables (which can be deduced in the applied case from the one for the two variable case) together with known results about the strong monodromy conjecture in the two variable case. Here an amazing cancellation occurs so that possible counterexamples fail.

Strong monodromy conjecture for defining polynomials of reduced projective curves having only weighted homogeneous singularities

Abstract

Let be a defining polynomial of a reduced projective curve having only weighted homogeneous singularities. We show that the strong monodromy conjecture for follows rather easily from arxiv:1609.04801v11 using a formula of Denef and Loeser for the Newton-nondegenerate case with three variables (which can be deduced in the applied case from the one for the two variable case) together with known results about the strong monodromy conjecture in the two variable case. Here an amazing cancellation occurs so that possible counterexamples fail.
Paper Structure (3 sections, 6 theorems, 13 equations)

This paper contains 3 sections, 6 theorems, 13 equations.

Table of Contents

  1. Topological zeta function
  2. Newton-nondegenerate case
  3. Proof of Theorem \ref{['T3']}

Key Result

Theorem 1

If there is no nonzero vector field of degree $0$ annihilating $f$, then $-k/d$ is a root of the Bernstein-Sato polynomial $b_f(s)$ of $f$ for any integers $k$ in $[3,d]$.

Theorems & Definitions (10)

  • Theorem 1: wh
  • Theorem 2: wh
  • Theorem 3
  • Proposition 1
  • Remark 1.1
  • Theorem 2.1: DL
  • Remark 3.1
  • Remark 3.2
  • Proposition 3.3
  • proof